Answer:
To solve this problem, you can use the principles of conservation of momentum and conservation of kinetic energy for an elastic collision.
(a) For the projected particle:
1. Break down the initial velocity of the projected particle into its x and y components. Since it's at an angle of 50 degrees, the x-component is 400 m/s * cos(50°) and the y-component is 400 m/s * sin(50°).
2. The x-component of momentum before the collision is the mass of the projected particle times its x-component velocity.
3. Since the target particle is initially at rest, the x-component of momentum after the collision is the same as before.
4. The y-component of momentum before and after the collision remains the same, as there is no external force in that direction.
5. Use the conservation of kinetic energy to find the combined kinetic energy of the particles after the collision.
6. Use the Pythagorean theorem to find the speed of the projected particle after the collision.
(b) For the target particle:
1. Use the conservation of momentum in the y-direction to find the y-component of velocity after the collision.
2. Since the target particle was initially at rest, the x-component of velocity remains zero.
3. Calculate the speed of the target particle after the collision using the Pythagorean theorem with its x and y components of velocity.
The calculations involve trigonometric functions, momentum conservation equations, and kinetic energy conservation equations. It's recommended to use a calculator or a software tool capable of handling these calculations efficiently.